# KC6034 - Complex Analysis

## What will I learn on this module?

The module is designed for you to develop the principles, techniques and applications of Complex Analysis.

Outline Syllabus:

Complex numbers: Basic algebraic properties; vectors and moduli; exponential form; products and powers.

Functions of complex variable: Mappings; limits; continuity; derivatives; Cauchy-Riemann equations; analytic functions; harmonic functions; exponential function; logarithmic function; branches and derivatives of logarithms; trigonometric functions; hyperbolic functions.

Integrals: Contours; contour integrals; brunch cuts; Cauchy’s integral theorem; Cauchy integral formula; Liouville’s theorem; fundamental theorem of algebra; maximum modulus principle.

Series: Convergence of sequences; converges of series; Taylor series; Laurent series; integration and differentiation of power series; multiplication and division of power series.

Residues and poles: Isolated singular points; residues; Cauchy’s residue theorem; three types of isolated singular points; residues at poles; zeros of analytic functions; behaviour of functions near isolated singular points; applications of residues.

# How will I learn on this module?

The module will be delivered using a combination of lectures and problem study classes, which will complement the taught material and enable you to obtain help with problems arising. In general, each topic presented in the lecture will be followed by a problem set and a problem study class. You will receive guidance on an individual basis to enhance your knowledge and expertise.

You will be assessed by a formal examination (70%) at the end of the module and a written assignment (30%) at the end of the first semester. The examination will cover all aspects of the module and will assess your problem solving abilities when applied to new unseen problems.

Informal feedback on work in progress will be given continuously during the module. You will also receive formal assignment feedback at the end of the first semester and examination feedback at the end of the module.

# How will I be supported academically on this module?

Lectures and problem solving classes will be the main point of academic contact, providing you with a formal teaching environment for core learning. Outside formal scheduled teaching, you will be able to contact the module team (module tutor, year tutor, programme leader) either via email or the open door policy operated throughout the programme. Further academic support will be provided through technology-enhanced resources via the e-learning portal. You will also have the opportunity to give your feedback formally through periodic staff-student committees and directly to the module tutor at the end of the lecture.

# What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk
(Reading List service online guide for academic staff this containing contact details for the Reading List team – http://library.northumbria.ac.uk/readinglists)

# What will I be expected to achieve?

Knowledge & Understanding:
1. Perform algebraic manipulations with complex numbers and functions of complex variable. (see KU1 and KU2 from Programme Learning Outcomes).
2. Evaluate basic contour integrals in the complex plane. Use Cauchy’s residue theorem to evaluate certain types of definite and improper integrals occurring in real analysis and applied mathematics. (see KU2 and KU3 from Programme Learning Outcomes)

Intellectual / Professional skills & abilities:

3. Construct rigorous mathematical arguments to produce relatively complex calculations, understanding their effectiveness and range of applicability (see IPSA1 from Programme Learning Outcomes).
4. Discuss and critically evaluate analytical approaches to formulating challenging problems of real analysis using methods of Complex Analysis (see IPSA3 from Programme Learning Outcomes).

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Demonstrate the ability to manage time and resources effectively (see PVA4 from Programme Learning Outcomes).

# How will I be assessed?

SUMMATIVE
1. Written assignment (30%) – 1, 3, 4, 5
Examination (70%) – 1, 2, 3, 4, 5

FORMATIVE
problem-solving classes – 1, 2, 3, 4, 5

Feedback will take several forms, including verbal feedback after lectures, individual verbal and written feedback following written assignment, and written feedback on the exam.

None

None

# Module abstract

Complex Analysis is one of the classical branches of mathematics with numerous applications in physics and engineering. The module will enable you to formulate mathematical problems in the language of functions of a complex variable. It will also allow you to develop advanced problem solving skills. The module will consist of lectures and problem solving sessions (timetables as lectures). You will be assessed by a final examination designed to put forward your newly developed skills and techniques. You will receive constructive feedback during lectures and problem solving sessions throughout the year, and eLearning Portal will serve as a point of contact, information and discussion with the tutor. The newly learned concepts and acquired skills will advance your knowledge of mathematics and will enhance your future employability.

# What will I learn on this module?

The module is designed for you to develop the principles, techniques and applications of Complex Analysis.

Outline Syllabus:

Complex numbers: Basic algebraic properties; vectors and moduli; exponential form; products and powers.

Functions of complex variable: Mappings; limits; continuity; derivatives; Cauchy-Riemann equations; analytic functions; harmonic functions; exponential function; logarithmic function; branches and derivatives of logarithms; trigonometric functions; hyperbolic functions.

Integrals: Contours; contour integrals; brunch cuts; Cauchy’s integral theorem; Cauchy integral formula; Liouville’s theorem; fundamental theorem of algebra; maximum modulus principle.

Series: Convergence of sequences; converges of series; Taylor series; Laurent series; integration and differentiation of power series; multiplication and division of power series.

Residues and poles: Isolated singular points; residues; Cauchy’s residue theorem; three types of isolated singular points; residues at poles; zeros of analytic functions; behaviour of functions near isolated singular points; applications of residues.

### Course info

UCAS Code G101

Credits 20

Level of Study Undergraduate

Mode of Study 4 years full-time or 5 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2020

## Mathematics MMath (Hons)

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